The photon model and equations are derived through time-domain mutual energy current

Transcription

1 The photon model and equations are derived through time-domain mutual energy current Shuang-ren Zhao *, Kevin Yang, Kang Yang, Xingang Yang and Xintie Yang March 7, 2017 Abstract In this article the authors will build the model of photon in timedomain. Since photon is a very short time wave, the authors need to build it in the time domain. In this photon model, there is an emitter and an absorber. The emitter sends the retarded wave. The absorber sends advanced wave. Between the emitter and the absorber the mutual energy current is built through the combination of the retarded wave and the advanced wave. The mutual energy current can transfer the photon energy from the emitter to the absorber and hence the photon is nothing else but the mutual energy current. This energy transfer is built in 3D space, this allow the wave to go through any 3D structure for example the double slits. The authors have proved that in the empty space, the wave can be seen approximately as 1D wave and can transfer energy from one pointer to to another point without any wave function collapses. That is why the light can be seen as light line. That is why a photon can go through double slits to have the interference. The duality of photon can be explained using this photon model. The total energy transfer can be divided as self-energy transfer and the mutual energy transfer. It is possible the self-energy current transfer half the total energy and it also possible that the part of self-energy part has no contribution to the energy transferring of the photon. In the latter, the self-energy items is canceled by the advanced wave of the emitter current and the retarded wave of the absorber current or canceled by the returned waves. This return wave is still satisfy Maxwell equations or at least some time-reversed Maxwell equations. Furthermore, the authors found the photon should satisfy the Maxwell equations in microcosm. Energy can be transferred only by the mutual energy current. In this solution, the two items in the mutual energy current can just interpret the line or circle polarization or spin of the photon. The traditional concept of wave function collapse in quantum mechanics is not needed in the authors' photon model. The authors believe the concept of the traditional wave collapse is coursed by the misunderstanding about the energy current. Traditionally, there is only the * Shuang-ren Zhao is with imrecons, in London, Canada, president@imrecons.com. Xintie Yang is with Aviation Academy, Northwestern Polytechnical University, Xi'an, China 1

2 energy current based on Poynting vector which is always diverges from the source. For a diverged wave, hence, there is the requirement for the energy to collapse to its absorber. After knowing that the electromagnetic energy is actually transferred by the mutual energy current, which is a wave diverging in the beginning and converging in the end, then the wave function collapse is not needed. The concept energy is transferred by the mutual energy current can be extended from photon to any other particles for example electron. Electrons should have the similar mutual energy current to carry their energy from one place to another and do not need the wave function to collapse. 1 Introduction The Maxwell equations have two solutions one is retarded wave, another is advanced wave. Traditional electromagnetic theory thinks there exists only retarded waves. The absorber theory of Wheeler and Feynman in 1945 oers a photon model which contains an emitter and an absorber. Both the emitter and the absorber sends half retarded and half advanced wave [1, 2, 8]. J. Welch brought the theory of the advanced wave or potential to electromagnetic theory and introduced time reciprocity theorem [17]. J. Cramer built the transactional interpretation for quantum mechanics by applied the absorber theory [4, 5] in around In 1978 Wheeler introduced the delayed choice experiment, which strongly implies the existence of the advanced wave [7, 18]. The delayed choice experiment is further developed to the delayed choice quantum eraser experiment[6], and quantum entanglement ghost image and the ghost image clearly oers the advanced wave picture[3]. The rst author of this article has introduced the mutual energy theory in 1987 [9, 20, 19]. Later he noticed that in the mutual energy theorem, the receive antenna sends advanced wave [10] and began to apply it to the study of the photon and the other quantum particles[14, 15, 13]. The above studies are all in Fourier domain which is more suitable to the case of the continual waves. The authors know the photon is the very short time wave, hence decided to study it in the time-domain instead of in the Fourier domain. The goal of this article is to build a model for the photon with classical electromagnetic theory and nd the equations for the photon. Some one perhaps will argue that photon is electromagnetic eld it should satisfy Maxwell equations, or photo is a particle it should satisfy Schrodinger equations, why now nd some other equations? First the authors are looking the vector equations which photon should satisfy. These equations cannot be Schrodinger equation. Second the innite more photons become light or electromagnetic eld radiation which should satisfy Maxwell equations, hence Maxwell equations are a macrocosm eld. But in microcosm, only singular photon, it is not clear whether the Maxwell equations still works. Hence for the singular photon, perhaps it satises Maxwell equation or perhaps it does not. The authors try to nd these equations for photon to satisfy, from which, if Maxwell equations should be possible to be derived as a limit in Macrocosm. 2

3 2 The photon model of Wheeler and Feynman In the photon model of Wheeler and Feynman, there is the emitter and absorber which sends all a half retarded wave and a half advanced wave. The wave is 1-D wave which is a plane wave send along x direction. This wave like a wave transfers in a cylinder wave guide. For both emitter and the absorber, the retarded wave is sent to the positive direction along the x. The advanced wave is sent to the negative direction along x. Color red is drawn to express the retarded waves. Color blue is drown to express the advanced wave, please see the Figure 2. For the retarded wave the arrow is drawn into the same direction of the wave. For the advanced wave the arrow in the opposite direction of the wave (since the energy transfers in the opposite direction for the advanced wave). Hence the arrow is always drawn in the energy transferring direction. For the absorber, Wheeler and Feynman assume the retarded wave sent by absorber is just negative (or having a 180 degree of phase dierence) compare to the retarded wave sent from the emitter. The advanced wave sent from the emitter is just negative (or 180 degree of phase dierence) of the advanced wave sent by the absorber. It is seems Wheeler and Feynman assumed that all the retarded waves and the advanced waves can only be sent in one direction, ether left or wright but both. See Figure 1. Hence, In the regions I and III, all the waves are canceled. In the region II the retarded wave from emitter and the advanced wave from absorber reinforced. All this model looks ok and it is very success in cosmography, but it is dicult to be believe. First why the retarded wave is sent by the emitter to the positive direction and the advanced wave is sent to the negative direction? The wave should sends to all directions, in 1-dimension situation, it should send to the positive direction and send to the negative direction. Why the absorber sends retarded wave just with a minus sign so it can cancel the retarded wave of the emitter? It is same to the Emitter, why it can send an advanced wave with minus sign so it just can cancel the advanced wave of the absorber? 1-D model is too simple. The wave is actually send to all direction instead of a 1-D plane wave, hence we need to check whether this model can be applied to 3D situations. What happens if this model for 3D? Lack of a 3D model for photon, it is perhaps the real reasons that Wheeler and Feynman theory and all the following theories, for example the transactional interpretation of J. Cramer, cannot be accept as a mainstream of the photon model for interpretation of the quantum mechanics. The authors endorse the absorber theory of Wheeler and Feynman. In this article we will introduce a 3D time-domain electromagnetic theory which suits to the advanced wave and retarded wave to replace the 1-D photon model of Wheeler and Feynman. In this new theory the mutual energy current[9, 20, 19, 14, 15, 13] will play an important role. 3

4 Figure 1: The Wheeler and Feynman model. The emitter sends retarded wave to right shown as red arrow. The emitter sends advanced wave to the left shown as blue. It is drawn the arrow in the opposite direction to the advanced wave. The absorber sends advanced wave to the left shown as blue and sends retarded wave to the right shown as red. However, the retarded wave of the absorber is just the negative value of the retarded wave (or it has 180 degree phase dierence) of the emitter. The advanced wave sent by the emitter is also with negative value (or has 180 degree phase dierence) of that of the absorber. Hence in the region I and III all the waves are canceled and in the region II, the waves are reinforced.. 4

5 3 The photon model of the authors The authors believe that in the traditional electromagnetic eld theory there is a mistakes to the understanding of the Poynting theorem and Lorentz reciprocity theorem. The energy current calculated by Poynting vector perhaps does not carry any energy in microcosm world like photons. Second the Lorentz reciprocity theorem actually is not a physic theorem but only mathematical transform of the mutual energy theorem, which is a real physic theorem in microcosm. The authors believe that the energy transfer for a singular photon from the emitter to the absorber can only be described with the mutual energy theorem. In microcosm, it is possible that the self-energy doesn't has any contribution to the energy transfer of a singular photon. Poynting theorem oers the theory about the self-energy hence the Poynting theorem is not important in microcosm. In this article it will be shown that in the macrocosm, the Poynting theorem can be derived from the mutual energy theorem which describes the energy transfer of a singular photon. In Fourier domain the mutual energy theorem and the Lorentz theorem can be derived from each others by applying a magnetic mirror transform. Hence there is the question that both the mutual energy theorem and the Lorentz theorem which is the original physic theorem, which is just a mathematical transform of a physic theorem? The author believe in empty space, the the mutual energy theorem is the original physic theorem. The Lorentz theorem is only a mathematical transform of the mutual energy theorem. In antenna calculation, it is never needed the concept of wave function collapse. The Lorentz theorem has been used to calculate the antenna problem which actually is because the Lorentz theorem contains the results of the mutual energy theorem. Since the photon can be seen as a system with an emitter and an absorber and can be further seen as a system with transmit antenna and a receive antenna. It is possible to apply the mutual energy theorem to the singular photon system. The rst author of this article has introduced the mutual energy theorem in 1987[9], found that the mutual energy current is just a inner product of two electric elds and pointed out that the mutual energy theorem is not just a transform of reciprocity theorem. The mutual energy theorem is established in lossless media, but the reciprocity theorem is established in symmetric media. The rst author of this article has applied the mutual energy theorem to spherical waves and plane waves [9, 20, 19]. The authors have proved that the mutual energy theorem can be derived from Poynting theorem and hence it is an energy theorem[10] and hence the concepts: self-energy, mutual energy, mutual energy current are all suitable. In that article it is also proved that the reciprocity can be derived from the mutual energy theorem. The authors also proved that in the lossy media, the mutual energy theorem is suitable but the the Lorentz theorem isn't[11]. Afterwords the authors begin to apply the mutual energy theorem to the photon model and quantum physics[12, 16, 15, 14]. However all the discussions are restricted to the eld in Fourier domain and the discussions only restricted to the mutual energy current but not the self-energy current. In 5

6 photon model the self-energy current is also very important. Is the self-energy current transfer energy? If it doesn't transfer energy, is it collapsed or canceled by some other? Or it just sends to the innity? In this article the authors will continue to prove that photon is nothing else but just the mutual energy current. The authors will show that the mutual energy current is never collapse and it is can be seen as 1D plane wave in a wave guide which has sharped tips at the two ends and which is very thick in the middle. The authors will prove this photon model theory based on the classical electromagnetic theorem with Maxwell equations. The authors will show that the Poynting theorem (and hence the self-energy current) in macrocosm is only a combination of many small mutual energy currents in microcosm. 3.1 The photon model The authors would like to nd the photon model from the solution of Maxwell equations. The photon should satises the following conditions. (I) The electromagnetic eld of the photon model should satises Maxwell equations. (II) The electromagnetic elds of the photon model should not be diverged. If the eld is diverged like the water waves, it need a concept of wave function collapse. We do not support the concept of wave function collapse, hence we seeks the electromagnetic solution which support converged wave that is the wave is allowed to be spread out in the beginning but it has to be converged in the end, so the absorber in one point can receive it without any wave function collapse. This wave somehow like the solitary particle. (III) The light sources or emitter should be possible to be put inside a metal container with a hole. The photon should be possible to send out from this hole. (I) The photon should be possible to go through not only a hole but double slits. () The macrocosm eld of point emitter should be the total contribution of the eld of many photons, which should satises the Poynting theorem and hence produce a diverged eld. In (II) we have said that the eld of microcosm can not be diverged, but the macrocosm eld of a emitter must be diverged like the water waves. (I) The eld of photon model has to support the polarization that means it must has two items and for the two items, the electric elds are perpendicular, and for two items it should be possible to allow a phase dierence 90 to 90 degrees. (II) The current of the emitter and the absorber must very simple, which have only one directions. Here it is not allowed the too complicated sources, for example like a antenna arrays. The current of the emitter and the absorber should be just have only one direction. (III) The force act on the photon must be consistent with energy conservation, momentum conservation. This condition is a little completed, it will not include in this article, but will be discussed in the future. 6

7 For photon model, the electric magnetic elds solution that satises the above conditions is looked by the authors. Next section the Poynting theorem is revised. 4 Poynting theorem For a photon, all the energy has been received by one absorber. It is not known whether or not a photon can fully satisfy the Maxwell equations. From the Maxwell equations it is known that the solution is the wave which should be sent to all directions, but it is clear that the photon is a energy package that send the energy from a pointer to another pointer. How to put the photon to the solutions of the Maxwell equations? However, the author believe the equations of photon should be the Maxwell equations or at least very close to the Maxwell equation, and further even in the microcosm the Maxwell equations can be deviated, but for the total elds or the eld of innite photons, which is the eld in macrocosm should still satisfy the Maxwell equations. Hence the eld in macrocosm should still satises the Poynting theorem. The Poynting theorem is following, ( E H ) ˆnd = ( J E + u) d (1) where E is the electric eld; H is the magnetic H-eld; J is the current intensity; is the boundary surface of volume ; u is the energy intensity saved on the volume ; ˆn is unit norm vector of the surface ; u is dened as u u t = E D t + H B t where D = ɛ E is the electric displacement; B = µb is magnetic B-eld; ɛ,µ are permittivity and permeability, which can be a scale value or a tensor; u is the increase of the energy intensity. The above equation is the Poynting theorem, which tells that the energy current comes through the surface to the inside the region! ( EÖ H ) ˆnd is equal to the energy loss ( J E )d in the volume and the increase of the energy inside the volume ( u)d. In this article energy current is equivalent to energy ux. Energy current sounds more real and energy ux sounds more virtual, the authors think the energy is real and hence the energy current is chosen instead of energy ux. But actually energy current is just the energy ux. It is known that form Poynting theorem all the reciprocity theorems can be derived[10]. It is also known that the Green function solution of the Maxwell equations can be derived from the reciprocity theorems. Hence, If all the solutions of the Maxwell equations can be derived, from principle, it should be possible do obtained Maxwell equations by induction. Hence even the Maxwell (2) 7

8 equation can not be derived from the Poynting theorem, but it is still can be said that the Poynting theorem contains nearly all the information of the Maxwell equations. Hence, if some eld satises Poynting theorem, it will be said that this eld almost satises the Maxwell equations. This point of view will be applied in the following sections of this article. 5 3D photon model in the time-domain with the mutual energy current Assume the electron in a atom can create some current which randomly sends half retarded wave and half advanced wave. Assume there are two this kind electrons 1 and 2, the energy lever of 1 is higher than 2. Assume that these two electrons send the wave just be synchronized which means that when the retarded wave of 1 reaches the 2, the 2 just sends the advanced wave, hence the advanced wave will reaches 1 at the the time the 1 sent the retarded wave. In this case 1 will be referred as emitter and 2 will be referred as absorber. The energy sent from 1 to 2 is the photon. All the retarded wave and advanced wave which are not synchronized can be all omitted because they have know contribution to the energy transfer from emitter to absorber. Assume the i-th photon is sent by an emitter 1 and received by an absorber 2. The current in the emitter can be written as J 1i, the current in the absorber can be written as J 2i. In the absorber theory of Wheeler and Feynman, the current is associated half retarded wave and half advanced wave. It is not taken here their choice in this moment, in the later of this article it will be discussed about this assumption. In this moment a very similar proposal is taken for the consideration. It is assumed that the emitter J 1i is associated only to a retarded wave and the absorber J 2i is associated only to an advanced wave. The photon should be the energy current sends from emitter to the absorber. For this proposal, it can be seen in Figure 1. It should be notice that in the following article there are two kinds of elds, one it the photon's eld which will have the subscript i, and this is the microcosm eld for example J 1i, E 1i, H 1i. Another is the eld without the subscript i, which is the macrocosm eld, for example J 1, E 1. Assume the advanced wave is existent same as retarded wave. Assume the current can be produced advanced wave and also retarded wave. In this case it is always possible to divide the current as two parts, one part created the advanced eld and the other part created the retarded wave. Assume J 1i produces retarded wave ξ 1i = [ E 1i, H 1i ]. J 2i produces advanced wave ξ 2i = [ E 2i, H 2i ]. Assume the total eld is a superimposed eld ξ i = ξ 1i + ξ 2i. Substitute ξ i = ξ 1i + ξ 2i and J i = J 1i + J 2i to Eq.(1). From Eq.(1) subtract the following self-energy items in the following, 8

9 which becomes ( E 1i Ö H 1i ) ˆnd = ( E 2i Ö H 1i ) ˆnd = ( J 1i E 1i + u 1i ) d (3) ( J 1i E 1i + u 1i ) d (4) ( E 1i H 2i + E 2i H 1i ) ˆnd = ( J 1i E 2i + J 2i E 1i ) d + ( E 1i D 2i + E 2i D 1i + H 2i B 1i + H 2i B 1i ) d (5) ( E 1i D 2i + E 2i D 1i + H 2i B 1i + H 2i B 1i )d is the increased mutual energy inside the volume. Assume the photon is sent from T = 0 and reached the absorber at t = T. Assume the photon is a short impulse with the time of t. Out side of the time window from the time t = 0 to the end t = T + t, this mutual energy should be vanishes. This part of the energy can be shown as the energy move from emitter to the absorber and in a particular time the energy is stayed at a place between the emitter and the absorber. Hence there is, ˆ ( E 1i D 2i + E 2i D 1i + H 2i B 1i + H 2i B 11 ) d dt = 0 (6) and hence there is ˆ ( E 1i Ö H 2i + E 2i Ö H 1i ) ˆnd dt = ˆ ( J 1i E 2i + J 2i E 1i ) d dt (7) If we call Eq.(3 and 4) as self-energy items of Poynting theorem, the Poynting theorem Eq.(1) with ξ = ξ 1i + ξ 2i are total eld of the Poynting theorem. Then the above formula Eq.(7) can be seen as mutual energy items of Poynting theorem. It also can be referred as mutual energy theorem because it is so important which will be seen in the following sections. Eq.(7) can be seen as the time domain mutual energy theorem. The selfenergy part of Poynting theorem Eq.(3, 4) perhaps is no sense. This is because that the self-energy current in the left side of Eq.(3, 4) cannot be received by any other substance. It can hit some atoms, but the atom has a very small section area and this self-energy current is diverged, so the self-energy received by the atom is so small and hence cannot produce a particle like a photon even 9

10 Figure 2: Photon model. There is an emitter and an absorber, emitter sends the retarded wave. The absorber sends the advanced wave. The photon is not vanishes at the time window between t = 0 and t = t + Δt. The photon has speed c. After a time T it travels to distance d = ct, where has an absorber. The gure shows in the time t = 1 2T, the photon is at the middle between the emitter and the absorber. The length of the photon is t c. The photon is shown with the yellow region. 10

11 with very long time. The authors do not accept the concept that the self-energy can collapse to some absorber. One of purpose of this article is to prove that without the concept of wave function collapse, the electromagnetic eld theory can still be possible to explain the phenomenon of the photon. Because photon is a particle, all its energy should eventually be received by the only one absorber. This part energy current (the self-energy item) is diverged and sent to innite empty space. Since we cannot accept a photon model, in which energy is continually lost, the self-energy current items in the side of the Eq.(3, 4), ether does not existent or need to be returned in late time. This two possibility will be discussed later in this article. For the moment we just ignore these two self-energy items. Assume all energy is transferred only through the mutual energy current items. We know that ξ 1i = [ E 1i, H 1i ] is retarded wave, ξ 2i = [ E 2i, H 2i ] is advanced wave. On the big sphere surface, ξ 1i is nonzero at a future time t = [T,T + t]. T = R c, where c is light speed, R is the distance from the emitter to the big sphere surface. t is the life time of the photon (from it begin to emit to it stop to emit, in which J 1i 0). Assume the distance between the emitter and the absorber is d with d R. J 2i 0 is at time [T,T + t], T = d c T. ξ 2i is an advanced wave and it is nonzero at [T T,T (T + t)] on the surface. Hence the following integral vanishes (ξ 1i and ξ 2i are not nonzero in the same time, on the surface ). One is in a time in the past and one is in a time in the future. In the above calculation we have assumed that T is very small compared with T, hence we can write T 0. Hence ξ 1 and ξ 2 are not nonzero at the same time in the the surface and hence there is, ( E 1i H 2i + E 2i H 1i ) ˆnd = 0 (8) The left side of Eq.(9) is the sucked energy by advanced wave E 2i from J 1i, which is the emitted energy of the emitter. ( J 2i E 1i ) d is the retarded wave E 1i act on the current J 2i. It is the received energy of J 2i. We can see that to prove the above surface integral vanishes is much easier in time domain compared to that in the Fourier domain[10], in the Fourier domain we have to prove all eld compounds cancel each other. This cancellation to prove the two elds just in the opposite directions is not needed in time domain. Notice that the above formula is very important, that means the mutual energy cannot be sent to the outside of our cosmos. In contrast for the selfenergy current, see the left side of Eq.(3, 4), there are energy current which is sent to the outside of our cosmos. The energy sends to the outside of our cosmos will be lost which doesn't meet the energy conservation law. Later we have to deal the problem of the self-energy current. In this moment we just omit the self-energy current and glad with the result that the mutual energy current does not sent to the outside of our cosmos. The above formula is only established when the ξ 1 and ξ 2 are one is retarded wave and another is advanced wave. If they are same wave for example both 11

12 are retarded waves the above formula Eq.(8) can not be established. This is also the reason we have to choose for our photon model as that one is retarded wave and the other is advanced wave. Hence from Eq.(v) and (8) we have ˆ ( J 1i E 2i ) d dt = ˆ ( J 2i E 1i ) d dt (9) In the above formula the left side is the emitted energy of the emitter, the right side is the absorbed energy of the absorber. The above formula tell us the emitted mutual energy is equal to the absorbed energy. Considering our readers perhaps are not all electric engineers, the authors make clear here why it is said that the left of the Eq.(9) is the emitted energy and the right side is the absorbed energy. In electrics, if there is an electric element with voltage U and current I, and they have the same direction, we obtained a power IU. This power is the loss of the energy of this electric element. If U has the dierent direction with current I or it has 180 degree phase dierence, this power is an output power to the system, i.e. this element actually is a power supply. In the case of the power consuming, IU = IU. In the case of power supply, the supplied power is IU = IU. Hence IU express a power supply to the system. Similarly, ( J 2i E 1i ) d is the power loss of absorber J 2i. ( J 1i E 2i ) d is the energy supply of J 1i. Assume 1 is a volume contains only the emitter J 1i. In this case, since there is a part of advanced wave and retarded wave and the two waves are synchronous. There is energy current go along the line linked the emitter and absorber. The energy current along the other paths not close to the line linked the emitter and absorber has dierent phase in wave and hence has only small contribution to the total energy transfer. Hence this part of energy current should not vanish, i.e., E 1i H 2i + E 2i H 1i ) ˆn d 0 (10) 1 ( See Figure 3. Eq.(7) can be rewritten as, ˆ J 1i E 2i ) d dt = ˆ ( ( 1 1 E 1i Ö H 2i + E 2i Ö H 1i ) ˆnddt (11) In this formula, 1 ( J 1i E 2i ) d is the emitted energy, this energy is the advanced wave E 2i sucked energy from the emitter current! J 1i. 1 ( E 1i 12

13 Figure 3: Red arrow is retarded wave, blue arrow is advanced wave. The direction of arrows show the directions of the energy currents. The emitter contains inside the volume 1. 1 is the boundary surface of 1. The mutual energy current consist of the retarded wave and the advanced wave can not vanish.. H 2i + E 2i H 1i ) ˆnd is the energy current from the emitter to the absorber, it is referred as the mutual energy current. In Figure 3, the red arrow is retarded wave, the blue arrow is the advanced wave. For retarded wave, the arrow direction is same as the wave direction. For the advanced wave the arrow direction is in the opposite direction of the wave. In the 3, we always draw the arrow in the energy current directions. For advance wave the energy current is at the opposite direction of the wave. Assume 2 is the volume which contains only the absorber J 2i, Eq.(5) can be written as ˆ 2 ( J 2i E 1 ) d dt = ˆ ( 2 Substitute Eq.(11 and 12) to Eq.(9) we obtain, = ˆ ˆ E 1i Ö H 2i + E 2i Ö H 1i ) ˆnddt (12) dt ( E 1i H 2i + E 2i H 1i ) ˆnd 1 dt ( E 1i H 2i + E 2i H 1i ) ( ˆn)d (13) 1 13

14 Figure 4: Choose the volume 2 is close to the absorber. Red arrows are retarded wave, blue arrows are advanced wave. The direction of retarded wave is same as the direction of red arrow. The direction of advanced wave is at the opposite direction of the blue arrow. The arrow direction (red or blue) is always at the energy transfer direction. The above formula tells us the all energy send out from 1 ows into (please notice the minus sign in the right) the surface 2. Consider the surface 1 and 2 is arbitrarily, that means in any surface between the emitter and the absorber has the same integral with same amount of the mutual energy current. Dene Q mi = E 1i H 2i + E 2i H 1i ) ˆnd (14) m ( here we change the direction of ˆn to be as always from the emitter to the absorber, then we can get the following formula, see Figure 5. ˆ t= Q mi dt = ˆ t= Q 1i dt m = 1, 2, 3, 4, 5 (15) From Figure 5 we can see that the time integral of the mutual energy current on an arbitrary surface are m are all the same, which are the energy transfer of the photon. In the place close to the emitter or absorber, the surface can be very small close to the size of the electron or atom. When energy is concentrated to a small region the momentum should also concentrated to that small region. In this case the energy of this mutual energy current will behaved like a particle. However, it is still the mutual energy current in 3D-space. Figure 5 and Eq.(15) tell us the energy transfer with the mutual energy current can be approximately seen as a 1-D plane wave i.e. a wave in a cylinder wave guide. The shape of the wave guide are with two sharp tips on the two ends of the wave guide and in 14

15 Figure 5: 5 surfaces are shown, the mutual energy current goes through each surface. The integrals of the mutual energy currents with time should be equal to each other in any of the above surface. the middle of the wave guide it become very thick. But the wave is actually as 3D wave, this allow the wave can go through the space other than empty space, for example double stilts. The mutual energy current has no any problem to go through the double slits and produce interference in the screen after the slits. This oers a clear interpretation for particle and wave duality of the photon. According the above discussion, the empty space can be used as a wave guide for the mutual energy current to transfer energy from the emitter to absorber. This kind of wave guide will be referred as the mutual energy wave guide. The mutual energy wave guide can be the whole empty space or just part of the empty space. For example if there is a wall between the emitter and the absorber and a hole on the wall, the mutual energy wave guide can not spread to the whole space and the hole restricts the thicknees of this kind of wave guide. In other hand if there are doubled slits on the wall, the wave guide can also be composed with the two paths through the two slits. 6 Self-energy items Which equations photon should satisfy? First we think the Maxwell equations. But it is seems that the Maxwell equations can only obtain the continual solution. But photon is a particle, its all energy is sent to an absorber direction instead sent to the whole directions. How can we obtained the solution of energy transfer from emitter to the absorber from the Maxwell equation? The eld of the solution should like solitary wave. Until now no one nd this kind wave can be surported by Maxwell equation, that is the reason the comcept of the wave function collapse has been introduced. Does photon satisfy Schrodinger wave 15

16 Figure 6: A photon in empty space, photon is just the mutual energy current. In some time, the photon is stayed at a place. This shows there is nothing about the concept of the wave function collapse. The mutual energy oers a 1-D cylinder wave guide which has two very sharp tips in the two ends. In the middle of the wave guide, it become very thick, it can be so thik even can occupy the whole space. equation? Schrodinger wave equation is scale equation, when there are many photons, the superimposed elds are electromagnetic elds and satisfy Maxwell equations. The elds are vector elds. Hence the eld of the photon should also be a vector eld and hence it cannot satisfy Schrodinger wave equation. We are interested to know which equations photon should satisfy. when the number of photon become innity, from these equations the Maxwell equations or Poynting theorem should be derived. In the photon model of last section, if only the mutual energy current has been considerred, everything is ne and there is no thing can be referred as the wave function collapse. However, in the Poynting theorem there are also self-energy items, in this section we need to oer a detail discussion about the self-energy items. In this section we need to consider the self-energy items Eq.(3 and 4). The advanced wave and retarded waves send to all directions instead send to only along the line linked the emitter and the absorber. In this model the absorber doesn't absorb all retarded wave of the emitter. The emitter doesn't absorb all advanced wave from absorber. The self-energy current of the wave in Eq.(3 and 4) is sent to innite space. The problem is what will happen for this self-energy current? 6.1 Self-energy current cannot vanish If there is only one source for example either an emitter or an absorber, the self-energy curren can not vanish. 16

17 First we have to know that the self-energy current cannot vanish. If selfenergy current vanishes, that from Poynting theorem Eq.(3) we can take a volume ab which is between two sphere surface a and b. In this volume there is no current hence we have, ( ( E 1i Ö H 1i ) ˆnd E 1i Ö H 1i ) ˆnd) = u 1i d (16) b ab a ( The self-energy current! b ( E 1i Ö H 1i ) ˆnd = 0 means the right side of the above formula is vanishes, that also means the left side of the above formula should vanish too. Hence we have ab u 1i d = 0 (17) Where ab is the volume between the two surface a and b, or or u 1i u 1i t = E 1i D 1i t + H 1i B 1i t = 0 (18) E 1i ɛ E 1i t = 0 (19) H 1i µ H 1i = 0 (20) t In the space the wave is nearly run in the direction as a line. In this situation E 1i (t) exp( jωt), E 1i t = jω E 1i (t). It is same to H 1i. Hence that the above equation require, E 1i E 1i = 0 (21) H 1i H 1i = 0 (22) That means the elds E 1i, H 1i must all vanish. It is same to E 2i, H 2i which should also vanish. The above discussion shows that if there is only a singular emitter or a sinular absorber, if the self enerngy current vanishes, their eld also vanishes. It sitll does not prove that in case there are emitter and absorber in the same time, the self-energy currents vanishes their elds also vanish. Even so, in this moment the self energy cannot vanish is assumed. We can think the eect of the self-energy is to help the mutual energy current to transfer the energy from the emitter to the absorber. After mutual energy current has nished its work, the energy of photon has been sent from the emitter to the absorber, the self-energy sill stayed in the space. What this self-energy 17

18 should do? If it doesn't return to the emitter or absorber, this energy will be continually lost to the outside of our universe. Our universe will continually loss energy that is also very strange. Hence we must think the possibility this energy can return to the emitter or the absorber. For self-energy current we can assume, (a) The self-energy items exist, they just send to innity. Because for the whole system with an emitter and an absorber there are one advanced wave and a retarded wave both send to innity, the pure total energy did not loss for the whole system. From the retarded wave the emitter loses some energy through the self-energy current, but from advanced wave the absorber lose the same amount of negative energy through the self-energy current. For the whole system including the emitter and the absorber, no energy is lost. The self energy is go from emitter to the absorber which loses some negtive energy means actually gains positive energy. The part of self-energy sent by the emitter can be seen as it is transferred to the innity. The self-energy sent by the absorber can be seen as some energy received from the innity. In this case the self-energy contributed to the energy transfer process of the photon. In the later it will be proved that the selfernergy currrent and the mutual energy current each contributes half of the energy transfer. (b) Because self-energy current cannot be absorbed by any things if it doesn't collapse. It need to collapse to a point to be absorbed. When mutual energy current can transfer energy, there is no any requirement for the wave to collapse. We can think the emitter sends an advanced wave and also sends a retarded wave which made the emitter doesn't lose or increase the energy through the self-energy. The absorber is also similar to the emitter, it sends also the advanced wave and also retarded wave. The absorber doesn't lose or increase the energy through the self-energy current. Hence, the self-energy items have no contribution to the energy transfer. (c) The self-energy current is existent. It helps the mutual energy to be sent from emitter to the absorber and hence the self-energy has come to the whole space. Afterwards, the self-energy return back to the emitter or absorber. Hence the self-energy current do not have any contribution to the energy transfer from emitter to the absorber. After the photon energy has been sent from emitter to the absorber through the mutual energy current, the energy sent by the emitter through self energy at the whole space returns back to the emitter and the energy sent by the absorber through self energy at the whole space returns back to the absorber. The reason we thought this part of self energy should return is that otherwise this part of energy will be lost to the outside of our universe and continuely losing the energy is violate the energy conservation. (d) It is same as (c), but the return eld of the self energy modied the original eld of self-energy, a new eld is produced which combine the original eld of the emitter or absorber and the return eld from the our universe. for the new eld the self-energy current is vanishes, but the eld itself doesn't vanish. This eld can also create the mutual energy current through the emitter and the absorber. 18

19 Figure 7: This shows emitters all at the center of the sphere. The absorbers are distributed at the surface of the sphere. The absorbers are the environment. We assume the absorbers are surrounded the emitters. This is our simplied macrocosm model. (e) The self energy current of the emitter collapse to the absorber. The self energy of the absorber collapse to the emitter in the same meaning as the quantum physics of Copenhagen. By the wave, here only the self-energy current is collapsed, the mutual energy current still not collopse. In the interpretation of Copenhagen, all wave is collopsed. The Figure 7 oers the authors' simplied macrocosm model. In this model the emitter is stay at the center of the sphere. In a big sphere there is many absorbers. This macrocosm model can be easily extended to more general situation where the emitter is not only stay at the center of the sphere but at a region close to the center. The absorber is also not only on the sphere but at all the place outside the sphere. (f) In order to to compare, the interpretation of Copenhagen also list here. In this situation, there is no advanced wave and the mutual energy all energy is transferred by self energy of the emitter through the retarded wave. 6.2 The idea of (a) First the idea of (a) is considered. The self-energy current which is retarded wave sent by the emitter is go to the innity. The self-energy current sent by the absorber is also go to the innity. This way the absorber obtained the energy which is equal to the lost energy of the emitter. Hence self-energy join the energy transfer. The energy is transferred not only by the mutual energy current but also by the self-energy current. We will prove in this situation the 19

20 self-energy current and the mutual energy current each has half contribution to the total energy transferring. If there is wave guide between the emitter to the absorber, the self-energy current is possible to transfer from the emitter to the absorber. The only dicult for this kind of energy transfer is that if there is metal container, and if the emitter and the absorber are all inside the container, how the self-energy current to be sent to the innity? We can think that the retarded self-energy current sends to the surface of metal container and become advanced wave of the absorber. But since the positions of emitter and absorber are in any places inside the container and the container can be any shape, there is no any electromagnetic theory can support this concept. Hence for this idea of (a) there is still some problem. However, in the following we will still continue working at the idea (a), and omit the situation of a metal container. We will prove that the Poynting theorem is satised in macrocosm for this situation. For idea (a) we can show even we throw away the self-energy items, it doesn't violate the Maxwell equations. After we throw away Eq.(3 and 4), there is only equation Eq.(5) left. We start from Eq.(5) to prove the Poynting theorem in macrocosm. Assume the emitters send retarded wave randomly with time. In the environment there are many absorbers in all directions which can absorb this waves. This is our simplied macrocosm model see Figure 7. Assume the self-energy doesn't vanish corresponding to the idea (a), we actually endorse the idea (c), but rst we check the idea (a), see if we don't worry about the situation in which the emitter and the absorber are all inside a metal container. We need to show that for (a) Maxwell equations are still satisfy for the macrocosm. Assume for the i-th photon the items of self-energy doesn't vanish, i.e., ( E 1i H 1i ) nd = ( J 1i E 1i + u 1i ) d (23) ( E 2i H 2i ) nd = ( J 2i E 2i + u 2i ) d (24) Assume for the i-th photon there is mutual energy current which satisfy: ( E 1i H 2i + E 2i H 1i ) ˆnd + = ( J 1i E 2i + J 2i E 1i ) d ( E 1i D 2i + E 2i D 1i + H 2i B 1i + H 2i B 1i ) d (25) 20

21 = ( i These 3 formulas actually tell us the photon should satisfy Poynting theorem, from the above equations can derive the Poynting theorem for the photon, ( E 1i + E 2i )Ö( H 1i + H 2i ) ˆnd + = ( J 1i + J 2i ) ( E 2i + E 1i ) d ( E 1i + E 2i ) ( D 1i + D 2i ) + ( H 1i + H 2i ) ( B 1i + B 2i ) d (26) Or we can take sum to the above formula it becomes, ( E 1i + E 2i ) ( H 1i + H 2i ) ˆnd i = ( J 1i + J 2i ) ( E 2i + E 1i )d i + ( E 1i + E 2i ) ( D 1i + D 2i ) + ( H 1i + H 2i ) ( B 1i + B 2i )d (27) i In another side, assume J 1 = i J 1i, J 2 = i J 2i, E 1 = i E 1i, E 2 = i E 2i, and so on. Hence there is, ( E 1 + E 2 )Ö( H 1 + H 2 ) E 1i + E 2j )Ö( H 1m + j m n H 2n ) E 1i m H 1m + i E 1i n H 2n + j E 2j m H 1m + j E 2j n H 2n = i (28) We have known the photon is a particle that means all energy of photon sends out from an emitter has to be received by only one absorber. Hence only the items with i = j doesn't vanish. Hence we have E 1i H 1m = E 1i H 1m = E 1i H 1i (29) i m im i In the above, considering E 1i H 1m = 0, if i m. This means that the eld of i-th absorber only act to i-th emitter. Similar to other items, hence we have ( E 1 + E 2 )Ö( H 1 + H 2 ) 21

22 = i ( E 1i H 1i + E 1i H 2i + E 2i H 1i + E 2i H 2i ) = i ( E 1i + E 2i ) ( H 1i + H 2i ) (30) And similarly we have, ( J 1 + J 2 )Ö( E 1 + E 2 ) = i ( J 1i + J 2i )Ö( E 1i + E 2i ) (31) (E 1 + E 2 ) (D 1 + D 2 ) = i (E 1i + E 2i ) (D 1i + D 2i ) (32) ( H 1 + H 2 ) ( B 1 + B 2 ) = i ( H 1i + H 2i ) ( B 1i + B 2i ) (33) Considering Eq.(30), Eq.(27) can be written as, ( E 1 + E 2 )Ö( H 1 + H 2 ) ˆnd + = ( J 1 + J 2 ) ( E 2 + E 1 )d ( E 1 + E 2 ) ( D 1 + D 2 ) + ( H 1 + H 2 ) ( B 1 + B 2 ) d (34) If we take = 1 which only contains the current of J 1 that means the current of environment J 2 is put out side of the volume 1, we have, ( E 1 + E 2 )Ö( H 1 + H 2 ) ˆnd = J 1 ( E 2 + E 1 )d 1 + ( E 1 + E 2 ) ( D 1 + D 2 ) + ( H 1 + H 2 ) ( B 1 + B 2 ) d (35) 1 Considering the total elds can be seen as the sum of the retarded wave and the advanced wave. In the macrocosm we do not know whether the eld is produced by the retarded eld of the emitter current J 1 or is produced by the advanced wave of the absorbers in the environment. We can think all the elds are produced by the source current J 1, hence we have E = E 1 + E 2, D = D 1 + D 2, H = H 1 + H 2, B = B 1 + B 2. Here the eld E, H are total electromagnetic eld in macrocosm, which are thought to be produced by emitter J 1, hence we have 22

23 = ( EÖH ) ˆnd 1 J 1 E d + E D + H B ) d (36) 1 ( This is the Poynting theorem in macrocosm. In this formula there is only emitter current J 1. The eld E, H can be seen as retarded wave but it is actually consist of both the retarded waves and the advanced waves in microcosm. We have started with assume the microcosm photon model where the eld is produced from the advanced wave of the absorber and the retarded wave of the emitter. We assume that the self-energy items doesn't vanish, we also assume there is the mutual energy current between the emitter and the absorber. All this means that for a singular photon it satises the Maxwell equations. We obtained the macrocosm Poynting theorem, in which the eld can be seen to be produced by the emitters. We know that Poynting theorem is nearly equivalent to the Maxwell equations. Although from Poynting theorem we cannot deduce Maxwell equations, but the Poynting theorem can derive all the reciprocity theorem[10], from the reciprocity theorem we can obtained the Green function solution of the Maxwell equations. From all solutions of Maxwell equations, the Maxwell equations should be possible to be induced from their all solutions. We have shown that if photon consists of the self-energy and the mutual energy items of an advanced wave and a retarded wave, the macrocosm electromagnetic eld which is summation of all elds of the emitters and absorbers still satisfy the Poynting theorem, and hence also the Maxwell equations in macrocosm. In our macrocosm model, the emitters are all at one point and all the absorbers are all on a sphere. However, this can be easily widened to more generalized situation in which the emitters are not only stay at one point and the absorbers are not only on a sphere surface but in the whole space. Perhaps the reader is still confuse here, and ask what are you doing? Started from Maxwell equation and prove the system satises the Maxwell equation? The reason is doing so is because we believe that for the eld of photon in microcosm, it doesn't require to satisfy the Maxwell equation exactly, but for the eld in macrocosm, it have to satisfy the Maxwell equations. The above derivation is that we believe in macrocosm, the Maxwell equation should be satised. If we cannot directly prove that the Maxwell equations is satised, at least the Poynting theorem should be satised in macrocosm. In microcosm which involves only the eld of a singular photon, this eld perhaps satises Maxwell equation, but perhaps not. In the above we have assumed the microcosm, the eld also satises the Maxwell equations and hence the Poynting theorem for self-energy and the mutual energy theorem. We also assume that the eld of the i-th emitter and the eld of j-th absorber do not contribute to energy or energy current, if i j, here i and j belong to dierent photon. This is also clear, the eld of dierent photons can not contributed to the energy current. We also assume that the macrocosm eld is the summation of all eld 23

24 of the emitter and also the absorber. The macrocosm eld includes all elds include retarded eld and also advanced eld in the microcosm. From all these assumptions we obtains that the macrocosm eld satises the Poynting theorem. From this we prove that this situation is a possible photon model. The only problem of the idea (a) is that if there is a metal container, how the self-energy current can be transferred to the innity? 6.3 For the idea of (b) In this situation all the self-energy current items don't transfer energy. The energy of photon is transferred only through the mutual energy items. We can assume the emitter also sends an advanced wave which have the same energy current as the retarded self-energy current, but has opposite direction of the energy transfer. Hence the total energy transfer of self-energy current for the emitter vanishes. We can assume for our universe the innite far away in the future is connected to the innite far away of the past. So the retarded self-energy current sent to the outside of our universe by the emitter will become the advanced wave and comeback to the emitter. It is similar to the absorber. The absorber has advanced self-energy current items. We assume the absorber also sends a retarded wave out which has the same amount of the energy current as the self energy current of the absorber and has a opposite direction. Hence the total self energy of the absorber also vanish. We assume the retarded wave and the advanced wave of the emitter is sent out in the time t = 0. We assume the advanced wave and the retarded wave of the absorber is sent out by the time of t = T = R c. Where R is the distance from the emitter to the absorber. c is the light speed. In this case, the retarded wave of the emitter and the advanced wave of the absorber can be synchronized. Hence the mutual energy current of the retarded wave of the emitter and the advanced wave of the absorber can produce nonzero mutual energy current. By the way, the retarded wave of the absorber begins at time t = T. When this wave reached to the emitter, it is time t = 2T. If the emitter sends the advanced wave also at t = 2T, these two waves can be synchronized. However, in the above when we speak about the emitter, it sends the retarded wave and also advanced wave, both waves are started at the time t = 0. The emitter sends the advanced wave at the same time as it sends the retarded wave. Hence, the retarded wave of the absorber and the advanced wave of the emitter cannot be synchronized and hence cannot produce any mutual energy current. There is only the mutual energy current between the retarded wave of the emitter and the advanced wave of the absorber. There is no any mutual energy current of the retarded wave of the absorber and the advanced wave of the emitter. Hence we have, ( E a 1iÖ H r 2i + E r 2iÖ H a 1i) nd = 0 24